23.4.174 problem 174
Internal
problem
ID
[6476]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
174
Date
solved
:
Tuesday, September 30, 2025 at 03:01:44 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 2 y y^{\prime \prime }&=8 y^{3}+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 57
ode:=2*y(x)*diff(diff(y(x),x),x) = 8*y(x)^3+diff(y(x),x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
\int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (4 \textit {\_a}^{2}+c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {1}{\sqrt {\textit {\_a} \left (4 \textit {\_a}^{2}+c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.871 (sec). Leaf size: 415
ode=2*y[x]*D[y[x],{x,2}] == 8*y[x]^3 + D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {-4 \text {$\#$1}^2}{-c_1}\right )}{\sqrt {4 \text {$\#$1}^2-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1-\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {-4 \text {$\#$1}^2}{-c_1}\right )}{\sqrt {4 \text {$\#$1}^2-c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {\text {$\#$1}} \sqrt {1+\frac {4 \text {$\#$1}^2}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {4 \text {$\#$1}^2}{c_1}\right )}{\sqrt {4 \text {$\#$1}^2+c_1}}\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-8*y(x)**3 + 2*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(2)*sqrt((-4*y(x)**2 + Derivative(y(x), (x, 2)))*y(x)) + De